Re: Speed of light straight from the classical wave equation ?

On Jan 30, 9:46 am, srp2...@xxxxxxxxx wrote:
On 30 jan, 09:14, Randy Poe <poespam-t...@xxxxxxxxx> wrote:

On Jan 30, 8:41 am, srp2...@xxxxxxxxx wrote:

Out of curiosity, I did a dimensional analysis of the very
straightforward wave equation
describing a moving transverse wave on an elastic string.

On the hyperphysics page,

v = sqrt[ T/(m/L) ]

Yes.

where m is mass of string, T is tension, and L is
length of string.

http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html

It would be more comprehensible to actually state
the equation you're talking about, or give a link.

The plane 2nd derivative equation you give below is fine.

From any intro ref on moving transverse waves on
solid strings.

you have mu which is in kg/m

Not necessarily, but I can imagine that there would
be an equation where the symbol mu was used to
represent mass per unit length.

It is in elementary intros to moving transverse wave on
elastic strings.

Also used in Griffiths intro to electrodynamics, which
is a popular ref.

A standard scalar wave equation is:

@^2 u/@t^2 = v^2 * del^2 u

The same, but with derivative on position at the left.

where "@" indicates partial derivative,

Ok thanks. Didn't know which symbol to use here.

u is the quantity
which is oscillating, del^2 means the Laplacian, and
the parameter v turns out to be the propagation
speed.

For EM radiation, u is a field strength. For
sound waves, u is pressure. For your oscillating
string, u is transverse distance. All of these
things have different units.

Different "macro" units, yes. But at the end,
they all simplify to velocity squared, of course.

you have T which is in Newton

you have (2nd p derivative on position) f = mu/T (2nd p derivative on
time) f

I take it f is the symbol for displacement here.

Yes. Transverse displacement



So mu/T = 1/v^2, or v = sqrt( T/mu ) as I said above.

Now, from E=mc^2 mass is really E/c^2, which is Joules/c^2

Yes, Joules/c^2 are a mass unit. I don't know if
I'd say mass is "really E/c^2". But that's one
possible system of units.

which gives mu the dimensions Joules/(c^2 m)

Newton is, after resolving to its most fundamental dimensions, Joules/
meter

What does "most fundamental" mean? Yes, a Newton
is a Joule/meter. It's also a kg*m/sec^2, which
is probably more fundamental since those are the
fundamental mks units.

Agreed, but you don't see the energy component then (joules).

which results in the dimensions of mu/T simplifying as follows

mu/T = [J/(c^2 m)] (m/J) = 1/c^2

Yes mu/T = 1/v^2 has the units of inverse velocity
squared.

More than that I think. If you introduce joules as a
unit, this 1/v^2 can be nothing but 1/c^2 by very definition
of mass as being E/c^2, directly from re-arranging E=mc^2.

Conclusion:

There is not even need to go to derivations from Faraday and
Maxwell's forth equation to get it.

No, there is no need to go to Faraday and Maxwell
to conclude that velocity of a wave on a string
has the units of velocity.

Not what I meant. I referred to getting c^2 from
the elementary wave equation.

Nor does anything in Faraday or Maxwell have anything
to say about what the velocity on a string is.

Nothing but the conclusion that by similarity with
the classical wave equation for moving wave on a
string, the product eps0 mu0 of the 2nd derivative
was equated to the inverse of a velocity squared,
which was the speed of light.

I sort of wonder what the meaning of this is.

I wonder what you think "this" is.

You seem to have concluded that velocity has
units of velocity (already known), that mu/T has
units of inverse velocity-squared (already known,
standard equation for speed of waves on a string),
and that you don't need Maxwell's equations to
recognize what parameter represents speed in a
general wave equation (also already known).

Known stuff of course, but not really my conclusion.

My conclusion is as stated. We can get c as a fixed
speed for energy from the straight classical wave
equation by simple dimensional analysis if we simplify
the standard units down to basic dimensions involving
energy.

But that's not what you calculated. You
calculated the DIMENSIONS of mu/T.

If you thought that this step

which results in the dimensions of mu/T simplifying as follows

mu/T = [J/(c^2 m)] (m/J) = 1/c^2

actually was saying that mu/T = 1/c^2 numerically
rather than dimensionally, you would have just
concluded that waves on a string propagate
at the speed of light.

That is untrue. As you started out saying, you
were just doing a dimensional analysis. Here's
how the numerical calculation would work out
in those units:

First we need to know how to convert between
cgs or mks units and E/c^2 mass units. An
electron has a mass of 511 keV/c^2, or
511e3 * 1.602e-19 = 8.19e-14 J/c^2. It also
has a mass of 9.11e-31 kg.

So 1 kg = 8.99e16 J/c^2.

Suppose I have a piece of string of mass 10 g
(0.01 kg, 8.99e14 J/c^2) and length 1 m. So
mu in these units is 8.99e14 (J/c^2)/m.

Suppose the tension on this string is 100 N,
or 100 J/m.

Then mu/T = (8.99e14 J/m.c^2) / (100 J/m)

= 8.99e12/c^2

As your dimensional analysis correctly determined,
mu/T has the dimensions of 1/c^2, i.e. it
can be expressed as a dimensionless constant
times 1/c^2.

Thus the speed of wave motion in this string,
sqrt(T/mu), is sqrt[c^2/8.99e12] = c/3e6,
which you can express in any units you like.
If we express c in m/sec, this gives us
that sqrt(T/mu) = 100 m/sec.

- Randy
.